Abstract
In this article, we are concerned with a nondifferentiable minimax fractional programming problem. We derive the sufficient condition for an optimal solution to the problem and then establish weak, strong, and strict converse duality theorems for the problem and its dual problem under B(p, r)invexity assumptions. Examples are given to show that B(p, r)invex functions are generalization of (p, r)invex and convex functions
AMS Subject Classification: 90C32; 90C46; 49J35.
Keywords:
nondifferentiable fractional programming; optimality conditions; B(p, r)invex function; duality theorems1 Introduction
The mathematical programming problem in which the objective function is a ratio of two numerical functions is called a fractional programming problem. Fractional programming is used in various fields of study. Most extensively, it is used in business and economic situations, mainly in the situations of deficit of financial resources. Fractional programming problems have arisen in multiobjective programming [1,2], game theory [3], and goal programming [4]. Problems of these type have been the subject of immense interest in the past few years.
The necessary and sufficient conditions for generalized minimax programming were first developed by Schmitendorf [5]. Tanimoto [6] applied these optimality conditions to define a dual problem and derived duality theorems. Bector and Bhatia [7] relaxed the convexity assumptions in the sufficient optimality condition in [5] and also employed the optimality conditions to construct several dual models which involve pseudoconvex and quasiconvex functions, and derived weak and strong duality theorems. Yadav and Mukhrjee [8] established the optimality conditions to construct the two dual problems and derived duality theorems for differentiable fractional minimax programming. Chandra and Kumar [9] pointed out that the formulation of Yadav and Mukhrjee [8] has some omissions and inconsistencies and they constructed two modified dual problems and proved duality theorems for differentiable fractional minimax programming.
Lai et al. [10] established necessary and sufficient optimality conditions for nondifferentiable minimax fractional problem with generalized convexity and applied these optimality conditions to construct a parametric dual model and also discussed duality theorems. Lai and Lee [11] obtained duality theorems for two parameterfree dual models of nondifferentiable minimax fractional problem involving generalized convexity assumptions.
Convexity plays an important role in deriving sufficient conditions and duality for nonlinear programming problems. Hanson [12] introduced the concept of invexity and established KarushKuhnTucker type sufficient optimality conditions for nonlinear programming problems. These functions were named invex by Craven [13]. Generalized invexity and duality for multiobjective programming problems are discussed in [14], and inseparable Hilbert spaces are studied by Soleimanidamaneh [15]. Soleimanidamaneh [16] provides a family of linear infinite problems or linear semiinfinite problems to characterize the optimality of nonlinear optimization problems. Recently, Antczak [17] proved optimality conditions for a class of generalized fractional minimax programming problems involving B(p, r)invexity functions and established duality theorems for various duality models.
In this article, we are motivated by Lai et al. [10], Lai and Lee [11], and Antczak [17] to discuss sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem with B(p, r)invexity. This article is organized as follows: In Section 2, we give some preliminaries. An example which is B(1, 1)invex but not (p, r)invex is exemplified. We also illustrate another example which (1, 1)invex but convex. In Section 3, we establish the sufficient optimality conditions. Duality results are presented in Section 4.
2 Notations and prelominaries
Definition 1. Let f : X → R (where X ⊆ R^{n}) be differentiable function, and let p, r be arbitrary real numbers. Then f is said to be (p, r)invex (strictly (p, r)invex) with respect to η at u ∈ X on X if there exists a function η : X × X → R^{n }such that, for all x ∈ X, the inequalities
hold.
Definition 2 [17]. The differentiable function f : X → R (where X ⊆ R^{n}) is said to be (strictly) B(p, r)invex with respect to η and b at u ∈ X on X if there exists a function η : X × X → R^{n }and a function b : X × X → R_{+ }such that, for all x ∈ X, the following inequalities
hold. f is said to be (strictly) B(p, r)invex with respect to η and b on X if it is B(p, r)invex with respect to same η and b at each u ∈ X on X.
Remark 1 [17]. It should be pointed out that the exponentials appearing on the righthand sides of the inequalities above are understood to be taken componentwise and 1 = (1, 1, ..., 1) ∈ R^{n}.
Example 1. Let X = [8.75, 9.15] ⊂ R. Consider the function f : X → R defined by
Let η : X × X → R be given by
To prove that f is (1, 1)invex, we have to show that
Now, consider
as can be seen form Figure 1.
Figure 1. φ = sin^{2}x + sin 2u(e^{12(1+u) } 1)  sin^{2}u.
Hence, f is (1, 1)invex.
Further, for x = 8.8 and u = 9.1, we have
Thus f is not convex function on X.
Example 2. Let X = [0.25, 0.45] ⊂ R. Consider the function f : X → R defined by
Let η : X × X → R and b : X × X → R_{+ }be given by
and
respectively.
The function f defined above is B(1, 1)invex as
as can be seen from Figure 2.
However, it is not (p, r) invex for all p, r ∈ (10^{17}, 10^{17}) as
(for x = 0.4 and u = 0.42)
< 0 as can be seen from Figure 3.
Hence f is B(1, 1)invex but not (p, r)invex.
In this article, we consider the following nondifferentiable minimax fractional programming problem:
(FP)
where Y is a compact subset of R^{m}, l(., .): R^{n }× R^{m }→ R, m(., .): R^{n }× R^{m }→ R, are C^{1 }functions on R^{n }× R^{m }and g(.): R^{n }→ R^{p }is C^{1 }function on R^{n}. D and E are n × n positive semidefinite matrices.
Let S = {x ∈ X : g(x) ≤ 0} denote the set of all feasible solutions of (FP).
Any point x ∈ S is called the feasible point of (FP). For each (x, y) ∈ R^{n }× R^{m}, we define
such that for each (x, y) ∈ S × Y,
For each x ∈ S, we define
where
Since l and m are continuously differentiable and Y is compact in R^{m}, it follows that for each x* ∈ S, Y (x*) ≠ ∅, and for any , we have a positive constant
2.1 Generalized Schwartz inequality
Let A be a positivesemidefinite matrix of order n. Then, for all, x, w ∈ R^{n},
Equality holds if for some λ ≥ 0,
If the functions l, g, and m in problem (FP) are continuously differentiable with respect to x ∈ R^{n}, then Lai et al. [10] derived the following necessary conditions for optimality of (FP).
Theorem 1 (Necessary conditions). If x* is a solution of (FP) satisfying x*^{T}Dx* >0, x*^{T}Ex* > 0, and ∇g_{h}(x*), h ∈ H(x*) are linearly independent, then there exist , k_{o }∈ R_{+}, w, v ∈ R^{n }and such that
Remark 2. All the theorems in this article will be proved only in the case when p ≠ 0, r ≠ 0. The proofs in the other cases are easier than in this one. It follows from the form of inequalities which are given in Definition 2. Moreover, without limiting the generality considerations, we shall assume that r > 0.
3 Sufficient conditions
Under smooth conditions, say, convexity and generalized convexity as well as differentiability, optimality conditions for these problems have been studied in the past few years. The intrinsic presence of nonsmoothness (the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and setvalued mappings) is one of the most characteristic features of modern variational analysis (see [18,19]). Recently, nonsmooth optimizations have been studied by some authors [2023]. The optimality conditions for approximate solutions in multiobjective optimization problems have been studied by Gao et al. [24] and for nondifferentiable multiobjective case by Kim et al. [25]. Now, we prove the sufficient condition for optimality of (FP) under the assumptions of B(p, r)invexity.
Theorem 2 (Sufficient condition). Let x* be a feasible solution of (FP) and there exist a positive integer s, 1 ≤ s ≤ n + 1, , , k_{o }∈ R_{+}, w, v ∈ R^{n }and satisfying the relations (2)(6). Assume that
(i) is B(p, r)invex at x* on S with respect to η and b satisfying b(x, x*) > 0 for all x ∈ S,
(ii) is B_{g}(p, r)invex at x* on S with respect to the same function η, and with respect to the function b_{g}, not necessarily, equal to b.
Then x* is an optimal solution of (FP).
Proof. Suppose to the contrary that x* is not an optimal solution of (FP). Then there exists an such that
We note that
Thus, we have
It follows that
From (1), (3), (5), (6) and (7), we obtain
It follows that
As is B(p, r)invex at x* on S with respect to η and b, we have
holds for all x ∈ S, and so for . Using (8) and together with the inequality above, we get
From the feasibility of together with , h ∈ H, we have
By B_{g}(p, r)invexity of at x* on S with respect to the same function η, and with respect to the function b_{g}, we have
Since b_{g}(x, x*) ≥ 0 for all x ∈ S then by (4) and (10), we obtain
By adding the inequalities (9) and (11), we have
which contradicts (2). Hence the result. □
4 Duality results
In this section, we consider the following dual to (FP):
where denotes the set of all satisfying
If, for a triplet , the set , then we define the supremum over it to be ∞. For convenience, we let
Let S_{FD }denote a set of all feasible solutions for problem (FD). Moreover, let S_{1 }denote
Now we derive the following weak, strong, and strict converse duality theorems.
Theorem 3 (Weak duality). Let x be a feasible solution of (P) and be a feasible of (FD). Let
(i) is B(p, r)invex at a on S ∪ S_{1 }with respect to η and b satisfying b(x, a) > 0,
(ii) is B_{g}(p, r)invex at a on S ∪ S_{1 }with respect to the same function η and with respect to the function b_{g}, not necessarily, equal to b.
Then,
Proof. Suppose to the contrary that
Then, we have
It follows from (5) that
with at least one strict inequality, since t = (t_{1}, t_{2}, ..., t_{s}) ≠ 0.
From (1), (13), (16) and (18), we have
Hence
Since is B(p, r)invex at a on S ∪ S_{1 }with respect to η and b, we have
From (19) and b(x, a) > 0 together with the inequality above, we get
Using the feasibility of x together with μ_{h }≥ 0, h ∈ H, we obtain
From hypothesis (ii), we have
As b_{g}(x, a) ≥ 0 then by (14) and (21), we obtain
Thus, by (20) and (22), we obtain the inequality
which contradicts (12). Hence (17) holds. □
Theorem 4 (Strong duality). Let x* be an optimal solution of (FP) and ∇g_{h}(x*), h ∈ H(x*) is linearly independent. Then there exist and such that is a feasible solution of (FD). Further, if the hypotheses of weak duality theorem are satisfied for all feasible solutions of (FD), then is an optimal solution of (FD), and the two objectives have the same optimal values.
Proof. If x* be an optimal solution of (FP) and ∇g_{h}(x*), h ∈ H(x*) is linearly independent, then by Theorem 1, there exist and such that is feasible for (FD) and problems (FP) and (FD) have the same objective values and
The optimality of this feasible solution for (FD) thus follows from Theorem 3. □
Theorem 5 (Strict converse duality). Let x* and be the optimal solutions of (FP) and (FD), respectively, and ∇g_{h}(x*), h ∈ H(x*) is linearly independent. Suppose that is strictly B(p, r)invex at a on S ∪ S_{1 }with respect to η and b satisfying b(x, a) > 0 for all x ∈ S. Furthermore, assume that is B_{g}(p, r)invex at a on S ∪ S_{1 }with respect to the same function η and with respect to the function b_{g}, but not necessarily, equal to the function b. Then , that is, is an optimal point in (FP) and
Proof. We shall assume that and reach a contradiction. From the strong duality theorem (Theorem 4), it follows that
By feasibility of x* together with μ_{h }≥ 0, h ∈ H, we obtain
By assumption, is B_{g}(p, r)invex at a on S ∪ S_{1 }with respect to η and with respect to the b_{g}. Then, by Definition 2, there exists a function b_{g }such that b_{g}(x, a) ≥ 0 for all x ∈ S and a ∈ S_{1}. Hence by (14) and (24),
Then, from Definition 2, we get
Therefore, by (25), we obtain the inequality
As is strictly B(p, r)invex with respect to η and b at on S ∪ S_{1}. Then, by the Definition of strictly B(p, r)invexity and from above inequality, it follows that
From the hypothesis , and the above inequality, we get
Therefore, by (13),
Since t_{i }≥ 0, i = 1, 2, ..., s, therefore there exists i* such that
Hence, we obtain the following inequality
which contradicts (23). Hence the results. □
5 Concluding remarks
It is not clear that whether duality in nondifferentiable minimax fractional programming with B(p, r)invexity can be further extended to secondorder case.
6 Competing interests
The authors declare that they have no competing interests.
7 Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
Izhar Ahmad thanks the King Fahd University of Petroleum and Minerals for the support under the Fast Track Project no. FT100023. Ravi P. Agarwal gratefully acknowledges the support provided by the King Fahd University of Petroleum and Minerals to carry out this research. The authors wish to thank the referees for their several valuable suggestions which have considerably improved the presentation of this article.
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